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Vol.36 (2-3) 2014 — 3 results found.

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Uniqueness of the Index Map in Banach Algebra K-Theory
C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (2-3) 2014, pp. 93–96
George A. Elliott (Received: 2014/06/18)

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It is shown that the index map in Banach algebra K-theory, as a natural map from the K\(_1\)-group of a quotient of a Banach algebra to the K\(_0\)-group of the corresponding ideal, is unique (up to an integral multiple).

Il est démontré que l’application index dans la K-théorie des algèbres de Banach est unique, dans un sens très naturel.

Quasitraces on Exact C*-algebras are Traces
C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (2-3) 2014, pp. 67–92
Uffe Haagerup (Received: 2011/03/13, Revised: 2014/03/24)

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It is shown that all 2-quasitraces on a unital exact \(C^*\)-algebra are traces. As consequences one gets: (1) Every stably finite exact unital \(C^*\)-algebra has a tracial state, and (2) if an \(AW^*\)-factor of type \(II_1\) is generated (as an \(AW^*\)-algebra) by an exact \(C^*\)-subalgebra, then it is a von Neumann \(II_1\)-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that \(RR(A)=0\) for every simple non-commutative torus of any dimension.

On démontre que toute 2-quasitrace sur une C*-algèbre exacte à élément unité est une trace. On en déduit les deux conséquences suivantes: (1) Toute C*-algèbre stablement finie et exacte possède un état tracial, et (2) si un AW*-facteur de type \(II_1\) est engendré (comme AW*-algèbre) par une sous-C*-algèbre exacte, il est une algèbre de von Neumann. Ceci est une solution partielle à un problème bien connu de Kaplansky. Le résultat principal a été utilisé par Blackadar, Kumjian, et Rørdam pour démontrer que \(RR(A) = 0\) pour tout tore non-commutatif simple de dimension quelconque.

A Classification of Tracially Approximate Splitting Interval Algebras~~I. The Building Blocks and the Limit Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (2-3) 2014, pp. 33–66
Zhuang Niu (Received: 2012/06/26, Revised: 2013/03/26)

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Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

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Most used Keywords

algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup cycles of ideals elliptic curves fixed point Fourier transform function fields. general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping noninterlacing numerical range orthogonal polynomials Predual space prime number property SP quadratic forms Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces Weak Markov set Whitney problems

Most used AMS

05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F20 37F25 39B72 42C05 43A07 43A62 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 60J75 83C05

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